Subgroups — C₂×C₇×#2847 #13213827

Order 1792.

8582 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#121	C₂	1	3	yes	maximal, complemented
3	C₂²×C₇×#120	C₂	1	2	yes	maximal, complemented
4	C₇×#2847	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#386	C₂	1	6	yes	maximal, complemented
6	C₂×C₇×#380	C₂	1	3	yes	maximal, complemented
7	C₂⁶×C₇	C₂²	1	1	yes	characteristic
8	C₂³×C₇×D₄	C₂²	1	6	yes	—
9	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
10	C₂⁴×C₄×C₇	—	2	24	no	—
11	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
12	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
13	C₂⁴×C₄×C₇	C₂²	1	8	yes	complemented
14	C₂×C₇×#121	—	2	24	no	—
15	C₂×C₇×#121	C₂²	1	24	yes	complemented
16	C₂×C₇×#120	—	2	16	no	—
17	C₂×C₇×#120	C₂²	1	16	yes	complemented
18	C₂²×C₇×C₄⋊C₄	C₂²	1	3	yes	complemented
19	C₂²×C₇×C₂²⋊C₄	C₂²	1	6	yes	—
20	C₂²×C₇×C₂²⋊C₄	C₂²	1	6	yes	complemented
21	C₇×#386	C₂²	1	48	yes	complemented
22	C₇×#380	C₂²	1	24	yes	complemented
23	C₂×C₇×#95	C₂²	1	6	yes	complemented
24	C₂×#2847	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₇	—	2	12	no	—
26	C₂⁵×C₇	—	2	12	no	—
27	C₂⁵×C₇	D₄	1	8	yes	complemented
28	C₂⁵×C₇	C₂³	1	2	yes	—
29	C₂⁵×C₇	C₂³	1	4	yes	—
30	C₂⁵×C₇	C₂³	1	3	yes	—
31	C₂²×C₇×D₄	—	2	24	no	—
32	C₂²×C₇×D₄	—	2	24	no	—
33	C₂²×C₇×D₄	—	2	24	no	—
34	C₂²×C₇×D₄	—	2	12	no	—
35	C₂²×C₇×D₄	C₂³	1	24	yes	—
36	C₂²×C₇×D₄	C₂³	1	4	yes	complemented
37	C₂³×C₄×C₇	—	2	128	no	—
38	C₂³×C₄×C₇	—	2	12	no	—
39	C₂³×C₄×C₇	—	2	24	no	—
40	C₂³×C₄×C₇	—	2	24	no	—
41	C₂³×C₄×C₇	—	2	96	no	—
42	C₂³×C₄×C₇	D₄	1	24	yes	complemented
43	C₂³×C₄×C₇	C₂³	1	12	yes	—
44	C₂³×C₄×C₇	C₂³	1	6	yes	—
45	C₂³×C₄×C₇	C₂³	1	3	yes	—
46	C₂³×C₄×C₇	C₂³	1	12	yes	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×#121	—	2	96	no	—
49	C₇×#120	—	2	64	no	—
50	C₂×C₇×C₄⋊C₄	—	2	12	no	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	complemented
52	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
53	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
54	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
55	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	complemented
56	C₇×#95	C₂³	1	24	yes	complemented
57	C₂⁵×C₄	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂²×#121	C₂×C₇	1	3	yes	complemented
59	C₂²×#120	C₂×C₇	1	2	yes	complemented
60	#2847	C₂×C₇	1	16	yes	direct product factor
61	C₂×#386	C₂×C₇	1	6	yes	complemented
62	C₂×#380	C₂×C₇	1	3	yes	complemented
63	C₂⁴×C₇	—	2	32	no	—
64	C₂⁴×C₇	—	2	96	no	—
65	C₂⁴×C₇	—	2	96	no	—
66	C₂⁴×C₇	—	2	24	no	—
67	C₂⁴×C₇	—	2	12	no	—
68	C₂⁴×C₇	—	2	24	no	—
69	C₂⁴×C₇	—	2	12	no	—
70	C₂⁴×C₇	—	2	12	no	—
71	C₂⁴×C₇	—	2	4	no	—
72	C₂⁴×C₇	—	2	12	no	—
73	C₂⁴×C₇	C₂×D₄	1	12	yes	—
74	C₂⁴×C₇	C₂×D₄	1	12	yes	—
75	C₂⁴×C₇	C₂×D₄	1	16	yes	complemented
76	C₂⁴×C₇	#41	1	12	yes	—
77	C₂⁴×C₇	C₂⁴	1	4	yes	—
78	C₂⁴×C₇	C₂⁴	1	6	yes	—
79	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₇×D₄	—	2	48	no	—
81	C₂×C₇×D₄	—	2	48	no	—
82	C₂×C₇×D₄	—	2	48	no	—
83	C₂×C₇×D₄	—	2	24	no	—
84	C₂×C₇×D₄	—	4	96	no	—
85	C₂×C₇×D₄	—	4	16	no	—
86	C₂²×C₄×C₇	—	2	96	no	—
87	C₂²×C₄×C₇	—	2	256	no	—
88	C₂²×C₄×C₇	—	2	24	no	—
89	C₂²×C₄×C₇	—	2	12	no	—
90	C₂²×C₄×C₇	—	2	96	no	—
91	C₂²×C₄×C₇	—	2	48	no	—
92	C₂²×C₄×C₇	—	2	48	no	—
93	C₂²×C₄×C₇	—	2	12	no	—
94	C₂²×C₄×C₇	—	2	12	no	—
95	C₂²×C₄×C₇	—	2	12	no	—
96	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
97	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
98	C₂²×C₄×C₇	C₂×D₄	1	48	yes	complemented
99	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
100	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
101	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
102	C₇×C₄⋊C₄	—	2	24	no	—
103	C₇×C₂²⋊C₄	—	2	48	no	—
104	C₇×C₂²⋊C₄	—	2	48	no	—
105	C₂⁶	C₂²×C₇	1	1	yes	characteristic
106	C₂³×D₄	C₂²×C₇	1	6	yes	—
107	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
108	C₂⁴×C₄	—	2	24	no	—
109	C₂⁴×C₄	C₂²×C₇	1	3	yes	—
110	C₂⁴×C₄	C₂²×C₇	1	3	yes	—
111	C₂⁴×C₄	C₂²×C₇	1	8	yes	complemented
112	C₂×#121	—	2	24	no	—
113	C₂×#121	C₂²×C₇	1	24	yes	complemented
114	C₂×#120	—	2	16	no	—
115	C₂×#120	C₂²×C₇	1	16	yes	complemented
116	C₂²×C₄⋊C₄	C₂²×C₇	1	3	yes	complemented
117	C₂²×C₂²⋊C₄	C₂²×C₇	1	6	yes	—
118	C₂²×C₂²⋊C₄	C₂²×C₇	1	6	yes	complemented
119	#386	C₂²×C₇	1	48	yes	complemented
120	#380	C₂²×C₇	1	24	yes	complemented
121	C₂×#95	C₂²×C₇	1	6	yes	complemented
122	C₂³×C₇	—	2	32	no	—
123	C₂³×C₇	—	2	256	no	—
124	C₂³×C₇	—	2	96	no	—
125	C₂³×C₇	—	2	24	no	—
126	C₂³×C₇	—	2	48	no	—
127	C₂³×C₇	—	2	96	no	—
128	C₂³×C₇	—	2	24	no	—
129	C₂³×C₇	—	2	48	no	—
130	C₂³×C₇	—	2	12	no	—
131	C₂³×C₇	—	2	12	no	—
132	C₂³×C₇	—	2	12	no	—
133	C₂³×C₇	—	2	4	no	—
134	C₂³×C₇	—	2	12	no	—
135	C₂³×C₇	—	4	8	no	—
136	C₂³×C₇	—	4	24	no	—
137	C₂³×C₇	—	4	24	no	—
138	C₂³×C₇	#121	1	48	yes	complemented
139	C₂³×C₇	C₂²×D₄	1	12	yes	—
140	C₂³×C₇	C₂²×D₄	1	12	yes	—
141	C₂³×C₇	C₂²×D₄	1	6	yes	—
142	C₂³×C₇	C₂²×D₄	1	2	yes	—
143	C₂³×C₇	C₂×#41	1	12	yes	—
144	C₂³×C₇	C₂×#41	1	6	yes	—
145	C₂³×C₇	C₂⁵	1	1	yes	characteristic
146	C₇×D₄	—	4	96	no	—
147	C₇×D₄	—	4	16	no	—
148	C₂×C₄×C₇	—	2	96	no	—
149	C₂×C₄×C₇	—	2	24	no	—
150	C₂×C₄×C₇	—	2	12	no	—
151	C₂×C₄×C₇	—	2	96	no	—
152	C₂×C₄×C₇	—	2	12	no	—
153	C₂×C₄×C₇	—	2	12	no	—
154	C₂×C₄×C₇	—	2	12	no	—
155	C₂×C₄×C₇	—	4	24	no	—
156	C₂×C₄×C₇	#120	1	16	yes	complemented
157	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
158	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
159	C₂⁵	—	2	12	no	—
160	C₂⁵	—	2	12	no	—
161	C₂⁵	C₇×D₄	1	8	yes	complemented
162	C₂⁵	C₂³×C₇	1	2	yes	—
163	C₂⁵	C₂³×C₇	1	4	yes	—
164	C₂⁵	C₂³×C₇	1	3	yes	—
165	C₂²×D₄	—	2	24	no	—
166	C₂²×D₄	—	2	24	no	—
167	C₂²×D₄	—	2	24	no	—
168	C₂²×D₄	—	2	12	no	—
169	C₂²×D₄	C₂³×C₇	1	24	yes	—
170	C₂²×D₄	C₂³×C₇	1	4	yes	complemented
171	C₂³×C₄	—	2	128	no	—
172	C₂³×C₄	—	2	12	no	—
173	C₂³×C₄	—	2	24	no	—
174	C₂³×C₄	—	2	24	no	—
175	C₂³×C₄	—	2	96	no	—
176	C₂³×C₄	C₇×D₄	1	24	yes	complemented
177	C₂³×C₄	C₂³×C₇	1	6	yes	—
178	C₂³×C₄	C₂³×C₇	1	12	yes	—
179	C₂³×C₄	C₂³×C₇	1	3	yes	—
180	C₂³×C₄	C₂³×C₇	1	12	yes	—
181	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
182	#121	—	2	96	no	—
183	#120	—	2	64	no	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	complemented
186	C₂×C₂²⋊C₄	—	2	24	no	—
187	C₂×C₂²⋊C₄	—	2	24	no	—
188	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
189	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	complemented
190	#95	C₂³×C₇	1	24	yes	complemented
191	C₂²×C₇	—	2	128	no	—
192	C₂²×C₇	—	2	96	no	—
193	C₂²×C₇	—	2	12	no	—
194	C₂²×C₇	—	2	24	no	—
195	C₂²×C₇	—	2	12	no	—
196	C₂²×C₇	—	2	24	no	—
197	C₂²×C₇	—	4	4	no	—
198	C₂²×C₇	—	4	12	no	—
199	C₂²×C₇	—	4	12	no	—
200	C₂²×C₇	—	8	16	no	—
201	C₂²×C₇	C₂×#120	1	4	yes	—
202	C₂²×C₇	C₂×#121	1	12	yes	—
203	C₂²×C₇	C₂×#121	1	24	yes	complemented
204	C₂²×C₇	#525	1	12	yes	—
205	C₂²×C₇	C₂³×D₄	1	3	yes	—
206	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
207	C₂²×C₇	C₂²×#41	1	3	yes	—
208	C₄×C₇	—	2	12	no	—
209	C₄×C₇	—	4	12	no	—
210	C₄×C₇	C₂×#120	1	8	yes	complemented
211	C₂⁴	—	2	32	no	—
212	C₂⁴	—	2	96	no	—
213	C₂⁴	—	2	96	no	—
214	C₂⁴	—	2	24	no	—
215	C₂⁴	—	2	12	no	—
216	C₂⁴	—	2	24	no	—
217	C₂⁴	—	2	12	no	—
218	C₂⁴	—	2	12	no	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	2	12	no	—
221	C₂⁴	C₂×C₇×D₄	1	12	yes	—
222	C₂⁴	C₂×C₇×D₄	1	12	yes	—
223	C₂⁴	C₂×C₇×D₄	1	16	yes	complemented
224	C₂⁴	C₇×#41	1	12	yes	—
225	C₂⁴	C₂⁴×C₇	1	4	yes	—
226	C₂⁴	C₂⁴×C₇	1	6	yes	—
227	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
228	C₂×D₄	—	2	48	no	—
229	C₂×D₄	—	2	48	no	—
230	C₂×D₄	—	2	48	no	—
231	C₂×D₄	—	2	24	no	—
232	C₂×D₄	—	4	96	no	—
233	C₂×D₄	—	4	16	no	—
234	C₂²×C₄	—	2	96	no	—
235	C₂²×C₄	—	2	256	no	—
236	C₂²×C₄	—	2	24	no	—
237	C₂²×C₄	—	2	12	no	—
238	C₂²×C₄	—	2	96	no	—
239	C₂²×C₄	—	2	48	no	—
240	C₂²×C₄	—	2	48	no	—
241	C₂²×C₄	—	2	12	no	—
242	C₂²×C₄	—	2	12	no	—
243	C₂²×C₄	—	2	12	no	—
244	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
245	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
246	C₂²×C₄	C₂×C₇×D₄	1	48	yes	complemented
247	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
248	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
249	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
250	C₄⋊C₄	—	2	24	no	—
251	C₂²⋊C₄	—	2	48	no	—
252	C₂²⋊C₄	—	2	48	no	—
253	C₂×C₇	—	2	24	no	—
254	C₂×C₇	—	8	4	no	—
255	C₂×C₇	#2847	1	8	yes	direct product factor
256	C₂×C₇	C₂²×#120	1	1	yes	characteristic
257	C₂×C₇	C₂²×#121	1	3	yes	—
258	C₂×C₇	C₂×#525	1	3	yes	—
259	C₂³	—	2	32	no	—
260	C₂³	—	2	96	no	—
261	C₂³	—	2	256	no	—
262	C₂³	—	2	12	no	—
263	C₂³	—	2	12	no	—
264	C₂³	—	2	24	no	—
265	C₂³	—	2	48	no	—
266	C₂³	—	2	96	no	—
267	C₂³	—	2	24	no	—
268	C₂³	—	2	48	no	—
269	C₂³	—	2	12	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	12	no	—
272	C₂³	—	4	8	no	—
273	C₂³	—	4	24	no	—
274	C₂³	—	4	24	no	—
275	C₂³	C₇×#121	1	48	yes	complemented
276	C₂³	C₂²×C₇×D₄	1	12	yes	—
277	C₂³	C₂²×C₇×D₄	1	12	yes	—
278	C₂³	C₂²×C₇×D₄	1	6	yes	—
279	C₂³	C₂²×C₇×D₄	1	2	yes	—
280	C₂³	C₂×C₇×#41	1	12	yes	—
281	C₂³	C₂×C₇×#41	1	6	yes	—
282	C₂³	C₂⁵×C₇	1	1	yes	characteristic
283	D₄	—	4	96	no	—
284	D₄	—	4	16	no	—
285	C₂×C₄	—	2	96	no	—
286	C₂×C₄	—	2	12	no	—
287	C₂×C₄	—	2	24	no	—
288	C₂×C₄	—	2	96	no	—
289	C₂×C₄	—	2	12	no	—
290	C₂×C₄	—	2	12	no	—
291	C₂×C₄	—	2	12	no	—
292	C₂×C₄	—	4	24	no	—
293	C₂×C₄	C₇×#120	1	16	yes	complemented
294	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
295	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
296	C₇	C₂×#2847	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	128	no	—
298	C₂²	—	2	12	no	—
299	C₂²	—	2	24	no	—
300	C₂²	—	2	12	no	—
301	C₂²	—	2	24	no	—
302	C₂²	—	2	96	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	12	no	—
305	C₂²	—	4	12	no	—
306	C₂²	—	8	16	no	—
307	C₂²	C₂×C₇×#120	1	4	yes	—
308	C₂²	C₂×C₇×#121	1	12	yes	—
309	C₂²	C₂×C₇×#121	1	24	yes	complemented
310	C₂²	C₇×#525	1	12	yes	—
311	C₂²	C₂³×C₇×D₄	1	3	yes	—
312	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
313	C₂²	C₂²×C₇×#41	1	3	yes	—
314	C₄	—	2	12	no	—
315	C₄	—	4	12	no	—
316	C₄	C₂×C₇×#120	1	8	yes	complemented
317	C₂	—	2	24	no	—
318	C₂	—	8	4	no	—
319	C₂	C₇×#2847	1	8	yes	direct product factor
320	C₂	C₂²×C₇×#120	1	1	yes	characteristic
321	C₂	C₂²×C₇×#121	1	3	yes	—
322	C₂	C₂×C₇×#525	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁵×C₄×C₇	C₂	semidirect
2. C₂²×C₇×#121	C₂	semidirect
3. C₂²×C₇×#120	C₂	semidirect
4. C₇×#2847	C₂	direct (N × H)
5. C₂×C₇×#386	C₂	semidirect
6. C₂×C₇×#380	C₂	semidirect
9. C₂³×C₇×D₄	C₂²	semidirect
13. C₂⁴×C₄×C₇	C₂²	semidirect
15. C₂×C₇×#121	C₂²	semidirect
17. C₂×C₇×#120	C₂²	semidirect
18. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
20. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
21. C₇×#386	C₂²	semidirect
22. C₇×#380	C₂²	semidirect
23. C₂×C₇×#95	C₂²	semidirect
24. C₂×#2847	C₇	direct (N × H)
27. C₂⁵×C₇	D₄	semidirect
36. C₂²×C₇×D₄	C₂³	semidirect
42. C₂³×C₄×C₇	D₄	semidirect
51. C₂×C₇×C₄⋊C₄	C₂³	semidirect
55. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
56. C₇×#95	C₂³	semidirect
57. C₂⁵×C₄	C₂×C₇	semidirect
58. C₂²×#121	C₂×C₇	semidirect
59. C₂²×#120	C₂×C₇	semidirect
60. #2847	C₂×C₇	direct (N × H)
61. C₂×#386	C₂×C₇	semidirect
62. C₂×#380	C₂×C₇	semidirect
75. C₂⁴×C₇	C₂×D₄	semidirect
98. C₂²×C₄×C₇	C₂×D₄	semidirect
107. C₂³×D₄	C₂²×C₇	semidirect
111. C₂⁴×C₄	C₂²×C₇	semidirect
113. C₂×#121	C₂²×C₇	semidirect
115. C₂×#120	C₂²×C₇	semidirect
116. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
118. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
119. #386	C₂²×C₇	semidirect
120. #380	C₂²×C₇	semidirect
121. C₂×#95	C₂²×C₇	semidirect
138. C₂³×C₇	#121	semidirect
156. C₂×C₄×C₇	#120	semidirect
161. C₂⁵	C₇×D₄	semidirect
170. C₂²×D₄	C₂³×C₇	semidirect
176. C₂³×C₄	C₇×D₄	semidirect
185. C₂×C₄⋊C₄	C₂³×C₇	semidirect
189. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
190. #95	C₂³×C₇	semidirect
203. C₂²×C₇	C₂×#121	semidirect
210. C₄×C₇	C₂×#120	semidirect
223. C₂⁴	C₂×C₇×D₄	semidirect
246. C₂²×C₄	C₂×C₇×D₄	semidirect
255. C₂×C₇	#2847	direct (N × H)
275. C₂³	C₇×#121	semidirect
293. C₂×C₄	C₇×#120	semidirect
296. C₇	C₂×#2847	direct (N × H)
309. C₂²	C₂×C₇×#121	semidirect
316. C₄	C₂×C₇×#120	semidirect
319. C₂	C₇×#2847	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×C₄×C₇	2
C₂²×C₇×#121	2
C₂²×C₇×#120	2
C₇×#2847	2
C₂×C₇×#386	2
C₂×C₇×#380	2
C₂×#2847	7

Subgroups — C2×C7×#2847 #13213827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2847 #13213827